A Few More Useful 8-valued Logics for Reasoning with Tetralattice EIGHT 4статья
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Аннотация:In their useful logic for a computer network Shramko and Wansing generalize
initial values of Belnap’s 4-valued logic to the set 16 to be the power-set of Belnap’s 4. This
generalization results in a very specific algebraic structure — the trilattice SIXTEEN3
with three orderings: information, truth and falsity. In this paper, a slightly different
way of generalization is presented. As a base for further generalization a set 3 is chosen,
where initial values are a — incoming data is asserted, d — incoming data is denied,
and u — incoming data is neither asserted nor denied, that corresponds to the answer
“don’t know”. In so doing, the power-set of 3, that is the set 8 is considered. It turns
out that there are not three but four orderings naturally defined on the set 8 that form
the tetralattice EIGHT4. Besides three ordering relations mentioned above it is an extra
uncertainty ordering. Quite predictably, the logics generated by a–order (truth order)
and d–order (falsity order) coincide with first-degree entailment. Finally logic with two
kinds of operations (a–connectives and d–connectives) and consequence relation defined
via a–ordering is considered. An adequate axiomatization for this logic is proposed.